Global Well-Posedness for a Coupled Modified KdV System

TL;DR

This paper establishes the sharp global well-posedness for the initial value problems of the mKdV equation and a coupled mKdV system using advanced energy methods, extending the understanding of these nonlinear dispersive equations.

Contribution

It introduces the use of the refined I-method with almost conserved quantities to prove global well-posedness for the coupled mKdV system, a novel application in this context.

Findings

Proved global well-posedness for mKdV and coupled mKdV systems.

Applied the second generation of the modified energy method.

Extended the I-method to coupled nonlinear dispersive equations.

Abstract

We prove the sharp global well-posedness results for the initial value problems (IVPs) associated to the modified Korteweg-de Vries (mKdV) equation and a system modeled by the coupled modified Korteweg-de Vries equations (mKdV-system). To obtain our results we use the second generation of the modified energy and almost conserved quantities, more precisely, the refined I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.